An extension principle for Einstein-Maxwell-charged scalar field system with finite Doppler exponent
Abstract
We prove a new extension principle for the spherically symmetric Einstein--Maxwell--charged scalar field system. Rather than imposing conditions for mass ratio 2m/r as in Christodoulou's small-mass-ratio criterion or as in An--Chen--Singh's proof [1] with 2m/r<3/8, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of α' with α'=∂vα/∂vr and α=∂v(rϕ)/∂vr. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.
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BibTeX
@misc{an2026extension,
title = {{An extension principle for Einstein-Maxwell-charged scalar field system with finite Doppler exponent}},
author = {Xinliang An and Haoyang Chen and Taoran He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20469},
eprint = {2610.20469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20469},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - An extension principle for Einstein-Maxwell-charged scalar field system with finite Doppler exponent
AU - Xinliang An
AU - Haoyang Chen
AU - Taoran He
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20469
UR - https://arxiv.org/abs/2610.20469
AB - We prove a new extension principle for the spherically symmetric Einstein--Maxwell--charged scalar field system. Rather than imposing conditions for mass ratio 2m/r as in Christodoulou's small-mass-ratio criterion or as in An--Chen--Singh's proof [1] with 2m/r<3/8, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of α' with α'=∂_vα/∂_vr and α=∂_v(rϕ)/∂_vr. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.
ER -